Show that if $A^{n}=I$ then $A$ is diagonalizable.

Question

Suppose $A$ is an $m \times m$ matrix which satisfies $A^{n}=1$ for some $n$, then why is $A$ necessarily diagonalizable.

Not sure if this is helpful, but here's my thinking so far: We know that $A$ satisfies $p(x)=x^{n}-1=(x-1)(x^{n-1}+\ldots+x+1)$. If $A=I$ it is clearly diagonalizable so we may assume that $A$ is a root of the other factor.

Edit: Actually, I'm a bit confused and not even sure if we can say that much. Since the ring of $m \times m$ matrices is not an integral domain, we can not conclude that if $A-I \not = 0$ then $(A^{n-1}+\ldots+A+I)=0$, correct?

Answer 1

Let $P(X):=X^n-1$. Assume we work on an algebraically closed field $\mathbb K$ of characteristic $0$. We have, since $P$ kills $A$, that the minimal polynomial of $A$ splits on $\mathbb K[X]$ and has distinct roots. We conclude by Theorem 4.11.

Answer 2

Davide showed what happens in an algebraically closed field of characteristic $0$.

If the field is not algebraically closed, the result is not true, for example

$$A=\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}$$

Satisfies $A^4=I$ but is not diagonalizable over $\mathbb R$, as it has complex eigenvalues.

Also $$A=\begin{pmatrix} 1& 1\\ 0 & 1 \end{pmatrix}$$

is not diagonalizable in an algebraically closed field of characteristic $2$, but $A^2=I_2$. Note that the reason why $A$ is not diagonalizable is simple: both eigenvalues are $1$, thus if $A$ is diagonalizable, $D=I$ and thus $A=PDP^{-1}=I$ contradiction.